Journal of Materials Processing Technology 95 (1999) 169±179 Rapid prototyping and manufacturing technology applied to the forming of spherical gear sets with skew axes Ying-Chien Tsaia,*, Wern-Kueir Jehngb,1 a Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, 80424, R.O.C Department of Industrial Engineering and Management, National Kaohsiung Institute of Technology, Kaohsiung, Taiwan, R.O.C b Received May 1998 Abstract This paper proposes a systematic method to design and synthesize spherical gear sets in space Based on the kinematics of spherical mechanisms, a parametric mathematical model for spherical gear sets with skew axes is presented and a computer solid-model for spherical gears is designed and formed Using the gear solid-model database, non-ambiguous data set descriptions of rapid prototyping and manufacturing (RP&M) machines have been generated A design example is presented to demonstrate RP&M procedures for the surface generating of a complex spherical gear set with skew axes The result of this work can be of crucial bene®t in research in new gear types and industrial development In this paper, important detail techniques are also proposed for the solid-models of CAD/CAM systems, describing how they can be converted successfully into the RP&M (SL) system, and built up gear set models # 1999 Elsevier Science S.A All rights reserved Keywords: Rapid prototyping and manufacturing (RP&M); Solid-model; Gear set with skew axes; Three-dimensional printing (3DP); Instantaneous skew axis (ISA) Introduction In kinematics, spur and helical gears can be classi®ed as plane mechanisms, whilst bevel gears are spherical mechanisms, and hypoid and worm gears can be deemed as spatial mechanisms Therefore, the study of bevel gears should be based on spherical mechanisms Bevel gears are essential to power transmission between two intersecting axes Various mathematical models of tooth geometry for bevel gears have been presented by many authors; Huston and Coy [1] initially presented a new approach to the surface geometry of ideal spiral-bevel gears Tsai and Chin [2] further developed the surface geometry of straight and spiral-bevel gears with parametric representations of the spherical involute and the involute spiraloid to study the fundamental geometrical characteristics Most of these models are immediately available for 3D computer solid-models to provide for the analysis of their characteristics and performance Huston *Corresponding author Tel.: +886-7-525-2100; fax: +886-7-525-2149 E-mail address: yctsai@mail.nsysu.edu.tw (Y.-C Tsai) Ph.D Candidate of Mechanical Engineering Department, National Sun Yat-Sen University, Kaohsiung, Taiwan et al [3] investigated spiral-bevel gears of circular-cut pro®les for their geometric characteristics, such as radius of curvature, speci®c sliding and relative curvature, which are important to the analysis of meshing kinematics, contact stress, fatigue life and lubrication Existing bevel gears are mostly of spherical inviolate tooth pro®les or of tooth pro®les generated by tools with straight cutting edges In contrast, there are also many types of tooth geometry that may be used to give bevel gears conjugate tooth pro®les However, for simplicity of manufacturing, spherical tooth bodies are usually cut into conical shapes In this paper, using rapid prototyping and manufacturing technology, two meshing spherical gear sets with skew axes are designed and presented The RP primitives provide actual full-size physical models that can be handled, analyzed, and used for further development Rapid prototyping and manufacturing (RP&M) technology was developed in the late 1970s and early 1980s In the USA, Hebert of M in Minneapolis, Hull of UVP (Ultra Violet Products Ins.) in California, and Kodama of the Nagoya Prefectural Research Institute in Japan [4] worked independently on rapid prototyping concepts based on selectively curing a surface layer of photopolymer and 0924-0136/99/$ ± see front matter # 1999 Elsevier Science S.A All rights reserved PII: S - ( 9 ) 0 - 170 Y.-C Tsai, W.-K Jehng / Journal of Materials Processing Technology 95 (1999) 169±179 building three-dimensional objects with successive layers of the polymer Both Herbert and Kodama had dif®culty maintaining ongoing support from their research organizations, and each of them stopped his work before proceeding to a commercial or product phase UVP continued to support Hull, who worked through numerous problems of implementing photopolymer part building until he developed a complete system that could build detailed parts automatically Hull coined the term stereolithography, or threedimensional printing [5] This system was patented in 1986, at which time Hull and Freed, jointly with the stockholders of UVP, formed 3D Systems, to develop commercial applications in three-dimensional printing A different RP&M technology, developed by DTM in USA is a process of fusing or sintering called the selective laser sintering (SLS) system [6] This process generates three-dimensional parts by fusing powdered thermoplastic materials with the heat from an infrared laser beam A thin layer of powdered thermoplastic material is evenly spread, by a roller, over the build region The pattern of the corresponding part cross-section is then ``drawn'' by the laser on the powder surface With amorphous materials, the laser heat causes powder particles to soften and bind to one another at their points of contact, forming a solid mass The third RP&M technology was developed at the Massachusetts Institute of Technology (MIT) and is called threedimensional printing (3DP) [7] The 3DP process can use a number of powder materials including those well-known to the investment casting industry for the production of shells: refractory powder, such as silica or alumina, combined with a liquid colloidal silica binder Three-dimensional parts and ceramic molds are fabricated by selectively applying binder to thin layers of powder, causing the particles of powder to stick together Each layer is formed by generating a thin coating of powder and then applying binder to it with an inkjet-like mechanism Layers are formed sequentially and adhere to one another, thus generating a 3D object in a manner similar to that of the other RP&M systems The 3DP apparatus includes a cylinder ®tted with a piston that can be lowered in small increments under computer control The piston is coated with a thin layer of powder supplied by a hopper Above the powder is an ink-jet-like mechanism that is supplied with binder and can move along both horizontal (X±Y) axes Small droplets of liquid are ejected downwards continuously from the nozzle toward the powder surface Unwanted droplets are skimmed before reaching the powder by electrically charging them at the nozzle and then de¯ecting them from the stream by applying a voltage to electrodes located below the nozzle The nozzle is moved across the powder surface in a raster scan whilst the electrical signals control the deposit of binder onto the powder in speci®c locations After the molds have been fabricated from ceramic powder within the cylinder, they are placed in a furnace to cure the binder and strengthen the mold Following this step, excess unbound power is removed and the object is then ready for use In summary, RP&M technology is clearly a technology with enormous potential to form complicated solid prototypes The materials used include liquid resin, fusing powdered thermoplastic materials, PVC, polycarbonate, investment wax, nylon, ABS, powdered metals, ceramics, refractory powder (such as silica or alumina), etc The layeradditive methods have both laser and non-laser point-bypoint fabrications The layer faceted attachment also uses layer-additive and layer-subtractive fabrications The technology also uses many types of laser beams, such as ultra violet (UV) laser, infrared laser, UVargon-ion laser, UV He± Cd radiation, CO2 laser, etc Development work is continuing in USA, Japan, Germany, Sweden, Israel, and other European countries [8±10] Parametric tooth surfaces of spherical gear sets with skew axes Consider the moving coordinate systems S1(X1, Y1, Z1) and S2( X2, Y2, Z2) to be rigidly connected to gear and gear (shown in Fig 1) that transform rotation between two skew axes L1 and L2, and angles of gear rotation 01 and 02 for gear and gear 2, respectively Fixed coordinate systems S(X, Y, Z) and SH (XH , YH, ZH ) coincide with S1(X1, Y1, Z1) and S2(X2, Y2, Z2) respectively The notation  is denoted as the twisting angle of these two rotating axes By PluÈcker line coordinate theory, the common perpendicular line of L1 and L2 can be evaluated The X-axis and X1-axis coincide with the rotating axis L1, whilst the XH -axis and X2-axis coincide with the rotating axis L2 The Z-axis and ZH -axis coincide with the common perpendicular line of axes L1 and L2 The speed ratio of gear to gear is m The notations of  and u are the position parameters of the instantaneous screw axes,  is the twisting angle, whilst u is the Z-axis direction offset distance relative to L1 and the coordinate system S(X, Y, Z) Fig shows the kinematic vector relations for a two gear set with contact at point P The relative velocity Vrel of the conjugate tooth pro®les at contact point P is the rotating velocity of the L2 axis relative to the L1 axis There can be described two trihedrons, one being coincident with point P, and the other with point Q The trihedron at P has unit vectors nsa, n3 and n4, whilst the trihedron at Q has unit vectors n2, n3 and n5 The vector nsa is the unit direction vector of the ISA (de®ned pitch line) The unit vectors n1 that lies on the plane zˆu is perpendicular to the unit vector nsa The unit vectors n3 and n5 construct a datum plane called the N-plane [11] that is a plane necessary to depict the complex motion of the contacting point P The unit vector n2 is the unit normal vector of the N-plane The vector n3 is the unit direction vector of the line QP (de®ned &) Fig shows the parameters –,  and û, which are the position parameters of the N-plane Tsai and Sung [13] presented the ruled surface of the tooth pro®le of gear in coordinate system S1(X1, Y1, Z1) as follows: Y.-C Tsai, W.-K Jehng / Journal of Materials Processing Technology 95 (1999) 169±179 171 Fig Spatial rotation between two crossed skew axes, the derivation of coordinates, the instantaneous screw axes, and the pitch vertical P Q – cos  À & cos  sin  r1 …Y &Y –Y 01 † ˆ R …– sin  ‡ & cos  cos † cos 01 ‡ …u ‡ & sin † sin 01 SX À…– sin  ‡ & cos  cos † sin 01 ‡ …u ‡ & sin † cos 01 The ruled surface of the tooth pro®les of gear in coordinate system S2(X2, Y2, Z2) is described as: (1) InEq (3),and&arefunctionof–and01,i.e.,ˆ(–,01)and &ˆ&(–,01) Tsai and Sung [11], however, only directly simpli- P Q – cos … À † À & cos  sin … À † r1 …aY &Y –Y 01 † ˆ R ‰– sin … À † ‡ & cos  cos … À †Šcos 02 ‡ …u À E ‡ & sin †sin 02 SX À‰– sin … À † ‡ & cos  cos … À †Šsin 02 ‡ …u À E ‡ & sin †cos 02 The constraint for the parametric conjugate tooth pro®les of a gear set with skew axes was proposed by Tsai and Sung [8] as  & sin  d& d01     !   d d d d& À À cos d– d01 d– d01 ‡ …– sin  cos  sin  ‡ & cos  sin  ‡ & sin  cos  sin    d& ‡ u sin  sin  cos  ‡ u cos  sin † d–   d ‡ …À– & sin  sin  sin  ‡ u & cos  sin  cos † d– ‡ – sin  sin  cos  À u cos  cos  cos  ˆ 0X (3) (2) ®ed the constraints as point contact types to suit spur and bevel gears According to Litvin [12], ``Mating gear surfaces will differentiate surfaces € two cases of tangency: (i) the interacting€ € and are in line contact at every instant, and €2 is the envelope to the family of surfaces that is€ generated by in the € and are in point coordinate system S2, and (ii) surfaces € € contactateveryinstant(thecontactof and islocalized).'' Therefore, it is inconsequential to simplify Eq (3) to point contact only, whereas the meshing constraint equation of the spurandbevelgearshasalreadybeende®nedbyChangandTsai [13,14] In this paper, a special analytical solution of Eq (3) is 172 Y.-C Tsai, W.-K Jehng / Journal of Materials Processing Technology 95 (1999) 169±179 Fig N-plane, unit vectors, relative velocity, pitch line and pitch vertical proposed to apply for general situations and it is logically deduced fromspatial mechanismto sphericalmechanism This solutioncanbeusedtodesignanytypeofsphericalgearsetwith skew axes After extensive use of the trial and modi®cation methods, a special analytic solution for Eq (3) has been found, as follows: d ˆ d01 u …sin  cot  cot  ‡ tan  cos  cot2  cos  & À sin  tan  tan  cos  À cot  sin  tan  – À sin  tan  cos † ‡ …sin  sin  tan2  tan  & ! ‡ sin  cos  cot  À sin  cos  cot  tan †   tan  ‡ cot  ‡ …cos  ‡ sin  cot  sin †Y (4)  cot  d& ˆ ‰u…sin  cot  cot  ‡ tan  cos  cot2  cos  d01 À sin  tan  tan  cos  À cot  sin  tan  À sin  tan  cos † ‡ –…sin  sin  tan2  tan  ‡ sin  cos  cot  À sin  cos  cot  tan †ŠY (5) d ˆ …cot2  À tan2 †Y d– & (6) d& ˆ cot …cot  À tan †X d– (7) Consequently, by use of Eqs (1), (2)±(7), any types of skew gear sets in a spatial mechanism can be logically constructed 2.1 Deduction of general spherical gear sets with skew axes Since the spherical mechanism is a degeneration of the spatial mechanism, by using the theoretical kinematics of spherical mechanisms as proposed by Chiang [15], the spherical curvatures and torsion relations can be de®ned in Frenet±Senet form as: Q P Q Q P tH gn À nb gn À b R nH S ˆ R Àkg t ‡ (g b S ˆ R À g t SY bH t n t À (g n P (8) where unit vectors t, n, and b are taken at the current points of the spherical curve, as shown in Fig 3, forming a Y.-C Tsai, W.-K Jehng / Journal of Materials Processing Technology 95 (1999) 169±179 173 axes must have their two rotating axes intersecting at the spherical original point, so that it can be deduced that the offset distance of the two rotating axes is zero This means that uˆ0 (refer Eqs (1) and (2)), and Fig shows the locational relationships of spherical gear sets with skew axes, where coordinates S(X, Y, Z) and SH (XH , YH, ZH ) coincide at the center point O of the spherical ball For analyzing the loci of the contact point, it is necessary to create a datum for the X±Y plane (as shown in Fig 4) The connecting line of PP1 is the action line along the normal direction, and point P is the intersection point of the two pitch spherical cones The notation ! is the radius expanding angle from contact point P to point P1, therefore, OP1 is the instantaneous screw axis (ISA) of the two rotating axes Then PP1 is de®ned as p, OP1 is de®ned as –, OP is the radius of the spherical ball, and the triangle OPP1 relationships are deduced: Fig Unit vectors of a moving trihedron at the current point of the spherical curve trihedron which indicates the curve direction and curvatures Normal curvature n is equal to 1/& in spherical mechanism &ˆr, where r is the radius of the assumed unit length, and nˆ1 g is geodesic curvature From the characteristics of spherical curvatures, curves of spherical bodies have uniform normal curvature, and spherical gear sets with skew r ˆ – sec !Y (9) & tan ! ˆ Y – (10) d& d! ˆ tan ! ‡ –sec2 ! Y d– d– (11) d& ˆ – d!X (12) Substituting uˆ0 and Eqs (9) and (10) into Eq (1), the tooth pro®le of gear in the coordinate system S1(X1, Y1, Z1) degenerates to Fig Spherical gear sets rotating between intersected skew axes 174 Y.-C Tsai, W.-K Jehng / Journal of Materials Processing Technology 95 (1999) 169±179 P Q cos ! cos  À sin ! cos  sin  r1 …rY 01 † ˆ r R …cos ! sin  ‡ sin ! cos  cos †cos 01 ‡ sin ! sin  sin 01 SX À…cos ! sin  ‡ sin ! cos  cos †sin 01 ‡ sin ! sin  cos 01 Similarly the tooth pro®les of gear in coordinate S2(X2, Y2, Z2) are (13) n5 ˆ …sin  sin  sin  À cos  cos †i À …cos  sin  sin  ‡ sin  cos †j ‡ …cos  sin †kX P Q cos ! cos … À † À sin ! cos  sin … À † r2 …rY 01 † ˆ r R ‰cos ! sin … À † ‡ sin ! cos  cos … À †Šcos 02 ‡ sin ! sin  sin 02 SX À‰cos ! sin … À † ‡ sin ! cos  cos … À †Šsin 02 ‡ sin ! sin  cos 02 From Fig 1, the position vector rL of the ISA in coordinate S(X, Y, Z) and the relative velocity VL of the points on rL only have components along the ISA, and therefore, they can be represented as: rL …–† ˆ –…cos  i ‡ sin  j† ‡ uk (15) V L ˆ 32  ‰rL …–† À EkŠ À 31  rL …–† (16) where 31 ˆ À31 iY 32 ˆ 32 cos  i ‡ 32 sin  j Due to the relative velocity VL, and the position vector rL of the ISA have the same direction, i.e., drL …–†  V L ˆ 0X d– (17) By setting m2 to be the speed ratio of gear to gear 1, i.e., letting m ˆ À…32 a31 †, Eq (17) can be deduced as [16]: sin  Y mˆ sin … À †   m sin  Y  ˆ tanÀ1 m cos  À uˆ mE…m À cos r† X m2 À 2m cos r ‡ (18) The relative velocity of gear with respect to gear 1, as well as the position vector of contact point rp, can be deduced as: V rel ˆ 32  …rp À Ek† À 31  rp Y (20) Amongst Eqs (18)±(20), only two equations are independent, i.e., Eqs (18) and (19) are the same equation Fig shows unit vectors nsa, n1, n2, n3, n4 and n5 as follows: (21) nsa ˆ cos  i ‡ sin  jY  à (22) n1 ˆ R%a2Yj nsa ˆ Àsin  i ‡ cos  jY P Q cos  sin  ‡ sin  sin  cos   Ã à n2 ˆ RYnsa RYn1 k ˆ R sin  sin  À cos  sin  cos  SY cos  cos  (23) ‡ …– sin  ‡ & cos  cos †j ‡ …u ‡ & sin †kX n3 ˆ RYnsa à P Q Àsin  cos  n1 ˆ R cos  cos  SY sin  n4 ˆ …sin  sin †i À …cos  sin †j ‡ cos  kY (24) (26) In spherical gear sets with skew axes, the two rotating axes intersect at the center point of the spherical ball, and therefore, the length of the common perpendicular line of the axes is zero, meaning that Eˆ0 Substituting Eˆ0 into Eq (19), therefore uˆ0 also Then Eq (25) and Eq (26) can be deduced as: V rel ˆ 32  rp À 31  rp Y (27) rp ˆ …– cos  À & sin  cos †i ‡ …– sin  ‡ & cos  cos †j (28) As n2 is the unit normal vector of the N-plane, and Vrel is normal to the line of &, Vrel is also normal to the N-plane Therefore, the vectors n2 and Vrel are mutually parallel vectors on the N-plane and may have different directions The mathematical relationship between n2 and Vrel can be represented as: V rel  n2 ˆ 0X (29) Substituting Eqs (23), (27) and (28) into Eq (29), and taking the k terms to be simpli®ed, Eq (29) can be deduced as: À &m sin  sin  cos… À † À &m cos  sin2  sin… À † ‡ & sin  cos  sin  ‡ & sin2  sin  cos  ˆ 0X (30) Then substituting Eq (17) into Eq (30) gives: À&m sin  sin  cos… À † À & cos  sin2  sin  ‡ & sin  cos  sin  ‡ & sin2  sin  cos  ˆ sin ‰À&m sin  cos … À † ‡ & sin  cos Š ˆ 0X  (25) rp ˆ uk ‡ –nsa ‡ &n3 ˆ …– cos  À & sin  cos †i ‡ & sin kX (19) (14) (31) However, only if sin ˆ0, Eq (31) is equal to zero That means that in spherical gear sets with skew axes, the twisting angle  of the N-plane (shown Fig 2) is 08 or 1808, and cos ˆ1 or À1 As mentioned above, in spherical gear sets with skew axes it can be assumed that ˆ08, uˆ0 and Eˆ0 Y.-C Tsai, W.-K Jehng / Journal of Materials Processing Technology 95 (1999) 169±179 Substituting these relations into Eq (3), the constraint governing equation degenerates to     d& d& ‡ – sin  sin  ˆ 0X (32) ‡ …& sin  sin † À d01 – Eq (32) is the constraint governing equation of spherical gear sets with skew axes From the geometric characteristics, substituting Eqs (10)±(12) into Eq (32) yields  ! d! d! ˆ sin  sin  ‡ tan ! tan ! ‡ – sec ! X (33) d01 d– Eq (33) is the constraint condition for gear 1: similarly it can be developed analogously for gear Therefore, the constraint condition for gear is  ! d! d! ˆ sin  sin … À † ‡ tan ! tan ! ‡ – sec ! X d02 d– (34) Eqs (33) and (34) are two quasi-linear partial differential equations of order unity, where the parameter  may be a constant, or may be a function of 01 In spur gear sets if  is constant, the pro®les of the gears are involutes, whereas if ˆ%/2‡3.501, the pro®les are cycloid In addition, when ˆ1.6581‡0.501, there is a circular-arc dedendum; when ˆ1.6581À0.501, there is a full addendum circular-arc [13] In this paper, it is assumed that the parameter  is a function of 01 (or 02) i.e., ˆ(01) (or ˆ(02)) The solution of Eqs (33) and (34) may be expressed as:  (35) ! ˆ sin  sin  …01 † d01 ‡ f1 …r†Y  (36) ! ˆ sin … À † sin  …02 † d02 ‡ f2 …r†X d! d ˆ ‰ f1 …r†ŠY dr dr d! ˆ sin … À † sin  …02 †Y d02 d! d ˆ ‰ f2 …r†ŠY dr dr Substituting Eq (12) into Eq (41), the constraint Eq (41) for gear becomes d! ˆ sin  sin X d01 Here,  ! ˆ sin  sin  d01 ‡ f1 …r†Y and  ˆ …01 †X (42) (43) Analogously for gear 2, yields:  ! ˆ sin … À † sin  d02 ‡ f2 …r†Y and  ˆ …02 †X (44) Furthermore, assuming that the pro®les of the gear sets are involutes, then the parameter  is a constant if the initial value of 01 and 02 are equal to zero Thus Eqs (43) and (44) yield: ! ˆ 01 sin  sin  ‡ f1 …r†Y (45) ! ˆ 02 sin  sin … À r† ‡ f2 …r†X (46) Then Eqs (45) and (46) can be used as the constraint governing equations to develop the spherical gear set with skew axes Technology of RP&M applied in forming spherical gear sets (38) The RP&M machines for three-dimensional duplicating machines are highly dependent on the electronic database input These systems take an electronic description of a three-dimensional object and reproduce that description into a solid object If the description is inadequate, the part generated will also be inadequate To avoid the phenomenon of ``garbage in and garbage out'', the geometric descriptions required for current RP&M equipment are provided by computer-aided design (CAD) systems Therefore, it is necessary to incorporate CAD into the RP&M environment, RP&M input ®les, data representation, part con®guration, etc [11] (39) 3.1 CAD system integration within an RP&M environment When the parameter ! is separately formulated as a function of r and 01 (or 02), i.e., !ˆ!(r, 01), !ˆ!(r, 02), then Eqs (35) and (36) can be derived as: d! ˆ sin  sin  …01 †Y d01 175 (37) (40) where …dadr†‰ f1 …r†Š ˆ …dadr†‰ f2 …r†Š Eqs (37) and (38) are the constraint equations for gear Eqs (39) and (40) are the constraint equations for gear Assuming that the contact type is point contact, and that the parameters of & and – are mutually orthogonal, they are always independent at any contacting instant so that …d&ad–† ˆ Eq (32) then degenerates to   d& ‡ – sin  sin  ˆ 0X (41) À d01 RP&M model data requirements that are proposed currently are non-ambiguous data descriptions of the part geometry to be generated Non-ambiguous data sets result in unique possible interpretations [17], so the model data must facilitate the generation of closed paths and differentiate between the ``inside'' and ``outside'' of the part Using cross-hatching algorithms can create vectors that solidify the area between the part boundaries and borders Problems will occur if the part geometry is not completely closed because either adjacent surface vertices may not connect or whole surfaces may be missing Therefore, the RP&M system software must have the ability to close gaps, otherwise 176 Y.-C Tsai, W.-K Jehng / Journal of Materials Processing Technology 95 (1999) 169±179 the system could leave an opening that would negatively affect part building because gaps in the borders will cause the hatching vectors to be incomplete or escape outside the part The boundary data of a part must also convey the orientation of the solid area Surface normal information pertaining to the object's boundary is used to indicate the orientation of the object's mass If this information is incorrect, walls with zero thickness, or twisted surfaces with con¯icting, impossible orientations that create a part called a ``MoÈbius Strip'' can occur [18] Therefore, an unambiguous geometrical description is necessary for the model In this paper, the authors use two kinds of solid-model CAD systems: one is Parametric Technologies' PRO/ENGINEER, and the other is SDRC I-DEAS A solid-model can be de®ned as a geometric representation of a bounded volume This volume is represented graphically via curves and surfaces, as well as non-graphically through a topological tree structure which provides a logical relationship that is inherent in solid-models [19] The topological data de®nes and maintains the connective relationships between the various faces and surfaces of geometry Each face of the object, including its normal orientation with respect to the objects mass, is maintained Therefore, solid-models, by de®nition, satisfy the requirements for RP&M input Most popular 3D solid-model CAD systems require translated processors to create the STL ®le for an RP&M system In PRO/E system, the users are allowed to create STL ®les Firstly, the user must specify the degree of resolution for the model curved surfaces by entering a quality value in the CAD system's interface The value can range from to 10, with 10 being the highest resolution, although higher resolution values generate larger STL ®le sizes Using a triangle range density higher than eight seems to give little improvement to the ®nal part A CAD system's internal accuracy also affects STL accuracy [20] The IDEAS system calls this accuracy the point coincidence value [19] This value is often determined in the system's default start-up ®le Some systems allow users to interactively change the model's accuracy With the PRO/E system, one accuracy value applies to all features within the part Therefore, geometries with relatively large and small radii with bene®t from using a tighter accuracy value PRO/E will regenerate all previous features using the new accuracy value [21] After creating STL part model ®les, the next step is to con®gure these ®les located in the space of the vat used to construct the part STL part ®les must always build supports to hold the part in place whilst it is being generated For the SL (stereolithography) system, supports are required for every part because SL is an additive process using a layer-by-layer approach occurring on the liquid resin surface., and the resin is ®lled in a large vat To properly constrain a given layer, it must be attached to the previous layer The initial layer is attached to the platform that always provides support Liquid-based RP&M systems must also be concerned with trapped volumes Trapped volumes are de®ned as spaces that hold liquid separate from the liquid in the vat These regions may require special recoating parameters that slow the build rate A change in orientation can often eliminate the trapped volumes For example, a cup right side up contains a trapped volume, but when turned upside down, it does not If simple reorientation is impractical, the CAD designer can drill (or cut) drain holes The holes can reduce the resin in the trapped volumes These holes can be plugged later, during the post-processing stage Using the above-mentioned detail technologies, perfect STL ®les are transferred from the CAD system to the RP&M's slice computer There, the RP&M operators can complete and form the prototypes of the CAD design parts An example of spherical gear sets with skew axes formed by RP&M Fig shows a spherical gear set with two twisting intersecting axes The set is con®gurated in a shell spherical ball, with an inner spherical radius of 105 mm and an outer spherical radius of 115.5 mm The radii are conjugated contacting in the spherical space The common perpendicular distance is zero, and the twisting angle of the N-plane as shown in Fig relative to the pitch line (the ISA) is zero This indicates that sin ˆ0 and cos ˆ1 The twisting angle of these two rotating axes is 16.2988, the speed ratio is À1.33 and the constant pressure angle is 27.4858 From Eqs (13), (14) and (42), the pro®le surfaces of the couple gear are presented in Figs and Fig shows the surface functions of Eq (13) rotating from 08 to 3608 along axis L1 whilst Fig shows the surface functions of Eq (14) rotating from 08 to 3608 along axis L2 Firstly, from Figs and 6, the useful gear surfaces must be searched and picked in rotating spherical space Obviously gear is assumed Fig Ruled curve contours of a spherical gear Y.-C Tsai, W.-K Jehng / Journal of Materials Processing Technology 95 (1999) 169±179 Fig Ruled curve contours of a spherical pinion Fig A completed gear computer solid-model by the moving copy method N1 …31 †average ˆ N2 …32 †average X Fig Two gearing teeth synthesized on the spherical gear shell blank as rotating counter clockwise 308 per tooth, whilst gear rotates clockwise 408 per tooth By the gear law of transmission: 177 (47) The speed ratio mˆÀ1.33 is given, so that 31/32ˆÀ40/ 30ˆÀ1.33 From Eq (47) if the number of teeth on gear is 16, then the number of teeth on the pinion is 12 From Fig 5, and the above-mentioned  angle, the parameter  is a constant and if the initial values of 01 and 02 are equal to zero, then the pro®le of gear takes the rotating angle from 08 to 308, and the pinion from À58 to À458 to avoid serious undercut when they are meshing The synthesis of one tooth of the gearing contour pro®les of the gear and pinion are shown in Fig The contour pro®les can be rotation copied by the PRO/E CAD system to complete the circular-teeth in the spherical ball shell space, as shown in Fig As mentioned in the above section on RP&M technologies, the gears couple may need to have some drain holes drilled for the inner trapped volumes and have supports built if they are formed by the SL type machines Fig shows the completed solid-models as built by PRO/E CAD systems By the postprocessing of the PRO/E systems, these models have proposed to the SLA translator to translate the data base of the PRO/E solid-model to STL ®les for RP&M machines Fig Supports and drain holes of computer sold gear models for RP and M machines 178 Y.-C Tsai, W.-K Jehng / Journal of Materials Processing Technology 95 (1999) 169±179 the solid-models of parts can have higher resolution, whilst the supports not The completed STL model ®les with triangle meshes are shown in Fig 10 These ®les can be transferred to RP&M machines to construct prototypes by SL machines as shown in Fig 11 Conclusion In this paper, the complicated meshing contour surfaces of spherical gear set with skew axes have been derived, their constraint governing equations have been systematically investigated, and the kinematic relationships between the relative velocity and the instantaneous screw axis have been presented This information is very useful for spherical gear design and manufacture The new technology of RP&M with the stereolithography method has been used to verify the completeness and correctness of the derived mathematical models for spherical gear sets Acknowledgements The authors are grateful for research support from the National Science Council of the ROC through contract no NSC-87-2212-E-110-010, as well as assistance in the forming of rapid prototyping models from the Aero Industry Development Company References Fig 10 The SLA models using triangle density created by PRO/E systems Fig 11 The RP and M prototypes for spherical gear sets with skew axes Entering the PRO/E SLA module, ®rstly the system must specify the resolutions of the solid-models and the supports To avoid excessively large STL ®les, as mentioned above, [1] R.L Huston, J.J Coy, Ideal spiral-bevel gears ± a new approach to surface geometry, J Mech Des., Trans ASME 103 (1981) 27± 113 [2] Y.C Tsai, P.C Chin, Surface geometry of straight and spiral-bevel gears, J Mech Transm Autom Des 109 (1987) 443±449 [3] R.L Huston, Y Lin, J.J Coy, Tooth profile analysis of circular-cut, spiral-bevel gears, J Mech Transm Autom Des 105 (1983) 132± 137 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